Miracle extensions: Difference between revisions

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Line 1: Line 1:
{{Breadcrumb|Miracle}}
The basic 7-limit [[miracle]] temperament has various [[extension]]s to the 11- and 13-limit. The following temperaments are discussed in this article:  
The basic 7-limit [[miracle]] temperament has various [[extension]]s to the 11- and 13-limit. The following temperaments are discussed in this article:  
* '''Miraculous''' (31 & 41) – tempering out 105/104, 144/143, 196/195, and 243/242;  
* '''Miraculous''' ({{nowrap|31 & 41}}) – tempering out 105/104, 144/143, 196/195, and 243/242;  
* '''Benediction''' (31 & 41f) – tempering out 225/224, 243/242, 351/350, and 385/384;  
* '''Benediction''' ({{nowrap|31 & 41f}}) – tempering out 225/224, 243/242, 351/350, and 385/384;  
* '''Manna''' (31f & 41f) – tempering out 225/224, 243/242, 325/324, and 385/384;  
* '''Manna''' ({{nowrap|31f & 41f}}) – tempering out 225/224, 243/242, 325/324, and 385/384;  


In addition, we also consider the only alternative 11-limit extension:  
In addition, we also consider the only alternative 11-limit extension:  
* '''Revelation''' (21 & 31) – tempering out 66/65, 99/98, 105/104, and 512/507.  
* '''Revelation''' ({{nowrap|21 & 31}}) – tempering out 66/65, 99/98, 105/104, and 512/507.  


As we will see in [[#Interval chain]], miraculous is the only extension whose complexity is at about the same level as the 11-limit. It is [[support]]ed by [[72edo|72f]]. The generator, representing 15/14, and 16/15, goes one step further to stand in for ~14/13, and you can find 11/9~16/13 just three generator steps away. Benediction and manna are available if we want to use the more accurately tuned [[patent val]] mapping of prime [[13/1|13]] in 72edo, in which they merge into one. However, benediction benefits from a flatter tuning such as [[103edo]] whereas manna benefits from a sharper tuning such as [[113edo]].  
As we will see in [[#Interval chain]], miraculous is the only extension whose complexity is at about the same level as the 11-limit. It is [[support]]ed by [[72edo|72f]]. The generator, representing 15/14, and 16/15, goes one step further to stand in for ~14/13, and you can find 11/9~16/13 just three generator steps away. Benediction and manna are available if we want to use the more accurately tuned [[patent val]] mapping of prime [[13/1|13]] in 72edo, in which they merge into one. However, benediction benefits from a flatter tuning such as [[103edo]] whereas manna benefits from a sharper tuning such as [[113edo]].  
Miraculous, benediction, and manna can all be extended to the 17-limit by recognizing 21/16~17/13, tempering out [[273/272]]. For miraculous it implies the generator also represents 17/16, which is supported by 72fg.


Another possible path which relates a sense of compromise is to temper out [[169/168]], leading to [[semimiracle]]. This has the effect of slicing the period in two, and is supported by [[62edo|62]], 72, and [[82edo|82]].  
Another possible path which relates a sense of compromise is to temper out [[169/168]], leading to [[semimiracle]]. This has the effect of slicing the period in two, and is supported by [[62edo|62]], 72, and [[82edo|82]].  
Line 23: Line 27:
|-
|-
! rowspan="2" | 11-limit
! rowspan="2" | 11-limit
! colspan="3" | 13-limit extensions
! colspan="3" | 17-limit extensions
|-
|-
! Miraculous
! Miraculous
Line 39: Line 43:
| 116.6
| 116.6
| 15/14, '''16/15'''
| 15/14, '''16/15'''
| 14/13
| 14/13, '''17/16'''
|  
|  
|  
|  
Line 46: Line 50:
| 233.3
| 233.3
| '''8/7'''
| '''8/7'''
| 15/13
| 15/13, 17/15
|  
|  
|  
|  
Line 53: Line 57:
| 349.9
| 349.9
| 11/9
| 11/9
| '''16/13'''
| '''16/13''', 17/14, 21/17
|  
|  
|  
|  
Line 60: Line 64:
| 466.6
| 466.6
| '''21/16'''
| '''21/16'''
| 13/10
| 13/10, 17/13
|  
| 17/13
|  
| 17/13
|-
|-
| 5
| 5
| 583.2
| 583.2
| 7/5
| 7/5
|  
| 24/17
|  
|  
|  
|  
Line 88: Line 92:
| 933.2
| 933.2
| 12/7
| 12/7
|  
| 17/10
|  
|  
|  
|  
Line 101: Line 105:
| 10
| 10
| 1166.5
| 1166.5
| 49/25, 55/28, 63/32, 88/45, 96/49, 108/55
| 49/25, 55/28, 63/32, <br>88/45, 96/49, 108/55
|  
| 39/20, 51/26, 77/39, <br>128/65, 135/68, 168/85
|  
| 51/26, 100/51
|  
| 51/26, 65/33
|-
|-
| 11
| 11
| 83.1
| 83.1
| 21/20, 22/21
| 21/20, 22/21
| 26/25
| 18/17, 26/25
|  
|  
|  
|  
Line 130: Line 134:
| 433.1
| 433.1
| 9/7
| 9/7
|  
| 22/17
|  
|  
|  
|  
Line 180: Line 184:
| 33/32, 36/35
| 33/32, 36/35
| 27/26
| 27/26
| 40/39
| 35/34, 40/39
|  
| 34/33
|-
|-
| 22
| 22
Line 194: Line 198:
| 33/28
| 33/28
|  
|  
|  
| 20/17
| 13/11
| 13/11
|-
|-
Line 223: Line 227:
|  
|  
| 20/13
| 20/13
|  
| 17/11
|-
|-
| 28
| 28
Line 229: Line 233:
| 33/20
| 33/20
|  
|  
|  
| 28/17
|  
|  
|-
|-
Line 236: Line 240:
| 44/25
| 44/25
|  
|  
|  
| 30/17
|  
|  
|-
|-
Line 243: Line 247:
| 66/35
| 66/35
|  
|  
|  
| '''32/17'''
|  
| 17/9
|-
|-
| 31
| 31
| 16.1
| 16.1
| 81/80, 99/98, 121/120
| 81/80, 99/98, 121/120
|  
| 66/65
| 105/104, 144/143, 196/195
| 105/104, 120/119, 136/135, <br>144/143, 154/153, 170/169
| 91/90
| 65/64, 78/77, <br>85/84, 91/90
|-
|-
| 32
| 32
Line 271: Line 275:
| 99/80
| 99/80
|  
|  
| '''16/13'''
| '''16/13''', 21/17
| 26/21
| 26/21
|-
|-
Line 285: Line 289:
| 99/70
| 99/70
|  
|  
|  
| 24/17
|  
| 17/12
|-
|-
| 37
| 37
Line 300: Line 304:
|  
|  
| 21/13
| 21/13
| '''13/8'''
| '''13/8''', 34/21
|-
|-
| 39
| 39
Line 320: Line 324:
| 99/50
| 99/50
|  
|  
| 180/91
| 77/39, 128/65, <br>168/85, 180/91
| 143/72, 195/98, 208/105
| 119/60, 143/72, 135/68, <br>153/77, 169/85, 195/98
|}
|}
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced


== Tunings ==
== Tunings ==
* 5-limit POTE: ~15/14 = 116.673
=== Prime-optimized tunings ===
* 7-limit POTE: ~15/14 = 116.675
* 5-limit POTE: ~16/15 = 116.673{{c}}
* 7-limit POTE: ~15/14 = 116.675{{c}}
* 11-limit POTE
* 11-limit POTE
** Miracle: ~15/14 = 116.633
** Miracle: ~15/14 = 116.633{{c}}
** Revelation: ~15/14 = 116.277
** Revelation: ~15/14 = 116.277{{c}}
* 13-limit POTE
* 13-limit POTE
** Miraculous: ~15/14 = 116.747
** Miraculous: ~15/14 = 116.747{{c}}
** Benediction: ~15/14 = 116.574
** Benediction: ~15/14 = 116.574{{c}}
** Manna: ~15/14 = 116.739
** Manna: ~15/14 = 116.739{{c}}
** Revelation: ~15/14 = 116.268
** Revelation: ~15/14 = 116.268{{c}}
 
=== Target tunings ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings (miracle)
|-
! Target
! Generator
! Eigenmonzo*
|-
| 5-odd-limit
| ~16/15 = 116.588{{c}}
| 5/3
|-
| 7-odd-limit
| ~15/14 = 116.588{{c}}
| 5/3
|-
| 9-odd-limit
| ~15/14 = 116.716{{c}}
| 9/5
|-
| 11-odd-limit
| ~15/14 = 116.716{{c}}
| 9/5
|}
 
{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Least squares tunings (miracle)
|-
! Target
! Generator
! Eigenmonzo*
|-
| 5-odd-limit
| ~16/15 = 116.578{{c}}
| {{Monzo| 0 -19 20 }}
|-
| 7-odd-limit
| ~15/14 = 116.573{{c}}
| {{Monzo| 0 -27 25 5 }}
|-
| 9-odd-limit
| ~15/14 = 116.721{{c}}
| {{Monzo| 0 117 -44 -19 }}
|-
| 11-odd-limit
| ~15/14 = 116.672{{c}}
| {{Monzo| 0 17 -11 -6 11 }}
|}
 
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings (miraculous)
|-
! Target
! Generator
! Eigenmonzo*
|-
| 13-odd-limit
| ~15/14 = 116.716{{c}}
| 9/5
|-
| 15-odd-limit
| ~15/14 = 116.993{{c}}
| 3/2
|}
 
{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Least squares tunings (miraculous)
|-
! Target
! Generator
! Eigenmonzo
|-
| 13-odd-limit
| ~15/14 = 116.846{{c}}
| {{Monzo| 0 141 -70 -35 84 -42 }}
|-
| 15-odd-limit
| ~15/14 = 116.820{{c}}
| {{Monzo| 0 127 -84 -36 100 -44 }}
|}
 
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings (benediction)
|-
! Target
! Generator
! Eigenmonzo*
|-
| 13-odd-limit
| ~15/14 = 116.595{{c}}
| 13/9
|-
| 15-odd-limit
| ~15/14 = 116.588{{c}}
| 5/3
|}
 
{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Least squares tunings (benediction)
|-
! Target
! Generator
! Eigenmonzo
|-
| 13-odd-limit
| ~15/14 = 116.56309{{c}}
| {{Monzo| 0 -234 39 4 -115 228 }}
|-
| 15-odd-limit
| ~15/14 = 116.56348{{c}}
| {{Monzo| 0 -251 22 5 -131 261 }}
|}
 
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings (manna)
|-
! Target
! Generator
! Eigenmonzo*
|-
| 13-odd-limit
| ~15/14 = 116.760{{c}}
| 13/10
|-
| 15-odd-limit
| ~15/14 = 116.725{{c}}
| 15/13
|}
 
{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Least squares tunings (manna)
|-
! Target
! Generator
! Eigenmonzo
|-
| 13-odd-limit
| ~15/14 = 116.780{{c}}
| {{Monzo| 0 18 -111 -76 43 204 }}
|-
| 15-odd-limit
| ~15/14 = 116.764{{c}}
| {{Monzo| 0 -37 -166 -77 59 243 }}
|}
 
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Minimax tunings (revelation)
|-
! Target
! Generator
! Eigenmonzo*
|-
| 11-odd-limit
| ~15/14 = 116.164{{c}}
| 11/9
|-
| 13-odd-limit
| ~15/14 = 116.164{{c}}
| 11/9
|-
| 15-odd-limit
| ~15/14 = 116.164{{c}}
| 11/9
|}
 
{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Least squares tunings (revelation)
|-
! Target
! Generator
! Eigenmonzo
|-
| 11-odd-limit
| ~15/14 = 116.198{{c}}
| {{Monzo| 0 -195 35 5 89 }}
|-
| 13-odd-limit
| ~15/14 = 116.249{{c}}
| {{Monzo| 0 -234 39 4 102 11 }}
|-
| 15-odd-limit
| ~15/14 = 116.229{{c}}
| {{Monzo| 0 -251 22 5 117 13 }}
|}


=== Tuning spectra ===
=== Tuning spectra ===
Line 342: Line 532:
|-
|-
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|
| 17/16
| 104.955
|
|-
|
| 17/15
| 108.343
|
|-
|-
|  
|  
| 15/8
| 15/8
| 111.731
| 111.731
|
|-
|
| 17/14
| 112.043
|  
|  
|-
|-
Line 354: Line 559:
| 13/10
| 13/10
| 113.553
| 113.553
|
|-
|
| 17/10
| 114.830
|  
|  
|-
|-
Line 364: Line 574:
| 11/9
| 11/9
| 115.803
| 115.803
|
|-
|
| 17/13
| 116.107
|  
|  
|-
|-
Line 369: Line 584:
|  
|  
| 116.129
| 116.129
|  
| Lower bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
Line 389: Line 604:
|  
|  
| 116.505
| 116.505
|  
| 103fg val
|-
|-
| 17\175
| 17\175
|  
|  
| 116.571
| 116.571
|  
| 175ffggg val
|-
|
| {{monzo| 0 -27 25 5 }}
| 116.573
| 7-odd-limit least squares
|-
|
| {{monzo| 0 -19 20 }}
| 116.578
| 5-odd-limit least squares
|-
|-
|  
|  
Line 434: Line 639:
|  
|  
| 116.667
| 116.667
|  
| 72fg val
|-
|
| {{monzo| 0 17 -11 -6 11 }}
| 116.672
| 11-odd-limit least squares
|-
|-
|  
|  
Line 445: Line 645:
| 116.716
| 116.716
| 9-, 11- and 13-odd-limit minimax
| 9-, 11- and 13-odd-limit minimax
|-
|
| {{monzo| 0 117 -44 -19 }}
| 116.721
| 9-odd-limit least squares
|-
|-
|  
|  
Line 455: Line 650:
| 116.755
| 116.755
|  
|  
|-
| 18\185
|
| 116.757
| 185cffggg val
|-
|-
|  
|  
Line 464: Line 664:
|  
|  
| 116.814
| 116.814
|  
| 113fgg val
|-
|
| {{monzo| 0 127 -84 -36 100 -44 }}
| 116.820
| 15-odd-limit least squares
|-
|
| {{monzo| 0 141 -70 -35 84 -42 }}
| 116.846
| 13-odd-limit least squares
|-
|-
|  
|  
Line 484: Line 674:
|  
|  
| 117.073
| 117.073
|  
| Upper bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
Line 494: Line 684:
| 13/9
| 13/9
| 117.559
| 117.559
|
|-
|
| 17/11
| 117.597
|  
|  
|-
|-
Line 499: Line 694:
| 13/12
| 13/12
| 117.936
| 117.936
|
|-
|
| 17/9
| 118.087
|
|-
|
| 17/12
| 119.400
|  
|  
|-
|-
Line 509: Line 714:
| 13/8
| 13/8
| 119.824
| 119.824
|
|-
|
| 21/17
| 121.942
|  
|  
|-
|-
Line 526: Line 736:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 543: Line 753:
| 11/9
| 11/9
| 115.803
| 115.803
|
|-
|
| 17/13
| 116.107
|  
|  
|-
|-
Line 548: Line 763:
|  
|  
| 116.129
| 116.129
|  
| Lower bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
Line 563: Line 778:
| 13/8
| 13/8
| 116.455
| 116.455
|
|-
|
| 17/16
| 116.501
|  
|  
|-
|-
Line 596: Line 816:
|-
|-
|  
|  
| {{monzo| 0 -234 39 4 -115 228 }}
| 17/14
| 116.56309
| 116.567
| 13-odd-limit least squares
|-
|  
|  
| {{monzo| 0 -251 22 5 -131 261 }}
| 116.56348
| 15-odd-limit least squares
|-
|-
| 17\175
| 17\175
|  
|  
| 116.571
| 116.571
| 175f val
|-
|
| 17/10
| 116.581
|  
|  
|-
|-
|  
|  
| {{monzo| 0 -27 25 5 }}
| 17/12
| 116.573
| 116.583
| 7-odd-limit least squares
|  
|-
|-
|  
|  
| {{monzo| 0 -19 20 }}
| 17/11
| 116.578
| 116.586
| 5-odd-limit least squares
|  
|-
|-
|  
|  
| 6/5
| 5/3
| 116.588
| 116.588
| 5-, 7- and 15-odd-limit minimax
| 5-, 7- and 15-odd-limit minimax
Line 655: Line 875:
|  
|  
|-
|-
| 7\72
|  
|  
| 116.667
| 17/9
| 116.642
|
|-
|
| 21/17
| 116.642
|
|-
|
| 17/15
| 116.666
|  
|  
|-
|-
| 7\72
|  
|  
| {{monzo| 0 17 -11 -6 11 }}
| 116.667
| 116.672
| Upper bound of 13- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
| 11-odd-limit least squares
|-
|-
|  
|  
Line 669: Line 899:
| 116.716
| 116.716
| 9- and 11-odd-limit minimax
| 9- and 11-odd-limit minimax
|-
|
| {{monzo| 0 117 -44 -19 }}
| 116.721
| 9-odd-limit least squares
|-
|-
|  
|  
Line 679: Line 904:
| 116.755
| 116.755
|  
|  
|-
| 18\185
|
| 116.757
| 185cfffgg val
|-
|-
|  
|  
Line 688: Line 918:
|  
|  
| 116.814
| 116.814
|  
| 113ffg val
|-
|-
|  
|  
Line 698: Line 928:
|  
|  
| 117.073
| 117.073
|  
| 41fg val, upper bound of 11-odd-limit diamond monotone
|-
|-
|  
|  
Line 710: Line 940:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 727: Line 957:
| 11/9
| 11/9
| 115.803
| 115.803
|
|-
|
| 17/13
| 116.107
|  
|  
|-
|-
Line 732: Line 967:
|  
|  
| 116.129
| 116.129
|  
| 31fg val, lower bound of 11-odd-limit diamond monotone
|-
|-
|  
|  
Line 752: Line 987:
|  
|  
| 116.505
| 116.505
|  
| 103ffgg val
|-
|-
| 17\175
| 17\175
|  
|  
| 116.571
| 116.571
|  
| 175fffgg val
|-
|
| {{monzo| 0 -27 25 5 }}
| 116.573
| 7-odd-limit least squares
|-
|
| {{monzo| 0 -19 20 }}
| 116.578
| 5-odd-limit least squares
|-
|-
|  
|  
Line 797: Line 1,022:
|  
|  
| 116.667
| 116.667
| Lower bound of 13- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|
| 17/15
| 116.667
|
|-
|
| 21/17
| 116.689
|  
|  
|-
|-
|  
|  
| {{monzo| 0 17 -11 -6 11 }}
| 17/9
| 116.672
| 116.702
| 11-odd-limit least squares
|
|-
|
| 17/10
| 116.707
|  
|-
|-
|  
|  
Line 808: Line 1,048:
| 116.716
| 116.716
| 9- and 11-odd-limit minimax
| 9- and 11-odd-limit minimax
|-
|
| {{monzo| 0 117 -44 -19 }}
| 116.721
| 9-odd-limit least squares
|-
|-
|  
|  
Line 818: Line 1,053:
| 116.725
| 116.725
| 15-odd-limit minimax
| 15-odd-limit minimax
|-
|
| 17/14
| 116.730
|
|-
|
| 17/12
| 116.750
|
|-
|-
|  
|  
Line 823: Line 1,068:
| 116.755
| 116.755
|  
|  
|-
| 18\185
|
| 116.757
| 185cf val
|-
|-
|  
|  
Line 830: Line 1,080:
|-
|-
|  
|  
| {{monzo| 0 -37 -166 -77 59 243 }}
| 17/16
| 116.764
| 116.785
| 15-odd-limit least squares
|-
|  
|  
| {{monzo| 0 18 -111 -76 43 204 }}
| 116.780
| 13-odd-limit least squares
|-
|-
|  
|  
Line 852: Line 1,097:
| 13/9
| 13/9
| 116.79299
| 116.79299
|
|-
|
| 17/11
| 116.801
|  
|  
|-
|-
Line 882: Line 1,132:
|  
|  
| 117.073
| 117.073
|  
| Upper bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
Line 894: Line 1,144:
|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)
! Unchanged interval<br>(eigenmonzo)
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 906: Line 1,156:
| 13/10
| 13/10
| 113.553
| 113.553
|
|-
| 2\21
|
| 114.286
|  
|  
|-
|-
Line 917: Line 1,172:
| 115.000
| 115.000
|  
|  
|-
| 5\52
|
| 115.385
| 52f val
|-
|-
|  
|  
Line 946: Line 1,206:
|  
|  
| 116.129
| 116.129
|  
| 11- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone (singleton)
|-
|-
|  
|  
Line 952: Line 1,212:
| 116.164
| 116.164
| 11-, 13- and 15-odd-limit minimax
| 11-, 13- and 15-odd-limit minimax
|-
|
| {{monzo| 0 -195 35 5 89 }}
| 116.198
| 11-odd-limit least squares
|-
|
| {{monzo| 0 -251 22 5 117 13 }}
| 116.229
| 15-odd-limit least squares
|-
|-
|  
|  
Line 967: Line 1,217:
| 116.241
| 116.241
|  
|  
|-
|
| {{monzo| 0 -234 39 4 102 11 }}
| 116.249
| 13-odd-limit least squares
|-
|-
|  
|  
Line 977: Line 1,222:
| 116.502
| 116.502
|  
|  
|-
| 10\103
|
| 116.505
|
|-
| 17\175
|
| 116.571
|
|-
|
| {{monzo| 0 -27 25 5 }}
| 116.573
| 7-odd-limit least squares
|-
|
| {{monzo| 0 -19 20 }}
| 116.578
| 5-odd-limit least squares
|-
|-
|  
|  
Line 1,011: Line 1,236:
|  
|  
| 116.667
| 116.667
|  
| 72ee val
|-
|-
|  
|  
Line 1,017: Line 1,242:
| 116.716
| 116.716
| 9-odd-limit minimax
| 9-odd-limit minimax
|-
|
| {{monzo| 0 117 -44 -19 }}
| 116.721
| 9-odd-limit least squares
|-
|-
|  
|  
| 9/7
| 9/7
| 116.792
| 116.792
|
|-
| 11\113
|
| 116.814
|  
|  
|-
|-
Line 1,041: Line 1,256:
|  
|  
| 117.073
| 117.073
|  
| 41ef val
|-
|-
|  
|  
Line 1,076: Line 1,291:
[[Category:Miracle]]
[[Category:Miracle]]
[[Category:Temperament extensions]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]

Latest revision as of 11:51, 6 August 2025

< Miracle

The basic 7-limit miracle temperament has various extensions to the 11- and 13-limit. The following temperaments are discussed in this article:

  • Miraculous (31 & 41) – tempering out 105/104, 144/143, 196/195, and 243/242;
  • Benediction (31 & 41f) – tempering out 225/224, 243/242, 351/350, and 385/384;
  • Manna (31f & 41f) – tempering out 225/224, 243/242, 325/324, and 385/384;

In addition, we also consider the only alternative 11-limit extension:

  • Revelation (21 & 31) – tempering out 66/65, 99/98, 105/104, and 512/507.

As we will see in #Interval chain, miraculous is the only extension whose complexity is at about the same level as the 11-limit. It is supported by 72f. The generator, representing 15/14, and 16/15, goes one step further to stand in for ~14/13, and you can find 11/9~16/13 just three generator steps away. Benediction and manna are available if we want to use the more accurately tuned patent val mapping of prime 13 in 72edo, in which they merge into one. However, benediction benefits from a flatter tuning such as 103edo whereas manna benefits from a sharper tuning such as 113edo.

Miraculous, benediction, and manna can all be extended to the 17-limit by recognizing 21/16~17/13, tempering out 273/272. For miraculous it implies the generator also represents 17/16, which is supported by 72fg.

Another possible path which relates a sense of compromise is to temper out 169/168, leading to semimiracle. This has the effect of slicing the period in two, and is supported by 62, 72, and 82.

For technical information see Gamelismic clan #Miracle.

Interval chain

In the following table, odd harmonics and subharmonics 1–21 are labeled in bold.

# Cents* Approximate ratios
11-limit 17-limit extensions
Miraculous Benediction Manna
0 0.0 1/1
1 116.6 15/14, 16/15 14/13, 17/16
2 233.3 8/7 15/13, 17/15
3 349.9 11/9 16/13, 17/14, 21/17
4 466.6 21/16 13/10, 17/13 17/13 17/13
5 583.2 7/5 24/17
6 699.9 3/2
7 816.5 8/5 21/13
8 933.2 12/7 17/10
9 1049.8 11/6 24/13
10 1166.5 49/25, 55/28, 63/32,
88/45, 96/49, 108/55 		39/20, 51/26, 77/39,
128/65, 135/68, 168/85 		51/26, 100/51 51/26, 65/33
11 83.1 21/20, 22/21 18/17, 26/25
12 199.8 9/8
13 316.4 6/5
14 433.1 9/7 22/17
15 549.7 11/8 18/13
16 666.3 22/15
17 783.0 11/7
18 899.6 27/16, 42/25 22/13
19 1016.3 9/5
20 1132.9 27/14, 48/25 52/27
21 49.6 33/32, 36/35 27/26 35/34, 40/39 34/33
22 166.2 11/10
23 282.9 33/28 20/17 13/11
24 399.5 44/35
25 516.2 27/20
26 632.8 36/25 13/9
27 749.5 54/35, 77/50 20/13 17/11
28 866.1 33/20 28/17
29 982.8 44/25 30/17
30 1099.4 66/35 32/17 17/9
31 16.1 81/80, 99/98, 121/120 66/65 105/104, 120/119, 136/135,
144/143, 154/153, 170/169 		65/64, 78/77,
85/84, 91/90
32 132.7 27/25 14/13 13/12
33 249.3 81/70 15/13 52/45
34 366.0 99/80 16/13, 21/17 26/21
35 482.6 33/25
36 599.3 99/70 24/17 17/12
37 715.9 121/80
38 832.6 121/75 21/13 13/8, 34/21
39 949.2 121/70 45/26 26/15
40 1065.9 231/125 24/13 13/7
41 1182.5 99/50 77/39, 128/65,
168/85, 180/91 		119/60, 143/72, 135/68,
153/77, 169/85, 195/98

* In 11-limit CWE tuning, octave reduced

Tunings

Prime-optimized tunings

  • 5-limit POTE: ~16/15 = 116.673 ¢
  • 7-limit POTE: ~15/14 = 116.675 ¢
  • 11-limit POTE
    • Miracle: ~15/14 = 116.633 ¢
    • Revelation: ~15/14 = 116.277 ¢
  • 13-limit POTE
    • Miraculous: ~15/14 = 116.747 ¢
    • Benediction: ~15/14 = 116.574 ¢
    • Manna: ~15/14 = 116.739 ¢
    • Revelation: ~15/14 = 116.268 ¢

Target tunings

Minimax tunings (miracle)
Target Generator Eigenmonzo*
5-odd-limit ~16/15 = 116.588 ¢ 5/3
7-odd-limit ~15/14 = 116.588 ¢ 5/3
9-odd-limit ~15/14 = 116.716 ¢ 9/5
11-odd-limit ~15/14 = 116.716 ¢ 9/5
Least squares tunings (miracle)
Target Generator Eigenmonzo*
5-odd-limit ~16/15 = 116.578 ¢ [0 -19 20⟩
7-odd-limit ~15/14 = 116.573 ¢ [0 -27 25 5⟩
9-odd-limit ~15/14 = 116.721 ¢ [0 117 -44 -19⟩
11-odd-limit ~15/14 = 116.672 ¢ [0 17 -11 -6 11⟩
Minimax tunings (miraculous)
Target Generator Eigenmonzo*
13-odd-limit ~15/14 = 116.716 ¢ 9/5
15-odd-limit ~15/14 = 116.993 ¢ 3/2
Least squares tunings (miraculous)
Target Generator Eigenmonzo
13-odd-limit ~15/14 = 116.846 ¢ [0 141 -70 -35 84 -42⟩
15-odd-limit ~15/14 = 116.820 ¢ [0 127 -84 -36 100 -44⟩
Minimax tunings (benediction)
Target Generator Eigenmonzo*
13-odd-limit ~15/14 = 116.595 ¢ 13/9
15-odd-limit ~15/14 = 116.588 ¢ 5/3
Least squares tunings (benediction)
Target Generator Eigenmonzo
13-odd-limit ~15/14 = 116.56309 ¢ [0 -234 39 4 -115 228⟩
15-odd-limit ~15/14 = 116.56348 ¢ [0 -251 22 5 -131 261⟩
Minimax tunings (manna)
Target Generator Eigenmonzo*
13-odd-limit ~15/14 = 116.760 ¢ 13/10
15-odd-limit ~15/14 = 116.725 ¢ 15/13
Least squares tunings (manna)
Target Generator Eigenmonzo
13-odd-limit ~15/14 = 116.780 ¢ [0 18 -111 -76 43 204⟩
15-odd-limit ~15/14 = 116.764 ¢ [0 -37 -166 -77 59 243⟩
Minimax tunings (revelation)
Target Generator Eigenmonzo*
11-odd-limit ~15/14 = 116.164 ¢ 11/9
13-odd-limit ~15/14 = 116.164 ¢ 11/9
15-odd-limit ~15/14 = 116.164 ¢ 11/9
Least squares tunings (revelation)
Target Generator Eigenmonzo
11-odd-limit ~15/14 = 116.198 ¢ [0 -195 35 5 89⟩
13-odd-limit ~15/14 = 116.249 ¢ [0 -234 39 4 102 11⟩
15-odd-limit ~15/14 = 116.229 ¢ [0 -251 22 5 117 13⟩

Tuning spectra

Miraculous

Edo
generator 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
17/16 104.955
17/15 108.343
15/8 111.731
17/14 112.043
13/10 113.553
17/10 114.830
7/4 115.587
11/9 115.803
17/13 116.107
3\31 116.129 Lower bound of 11- to 17-odd-limit,
and 17-limit 21-odd-limit diamond monotone
5/4 116.241
15/11 116.441
7/5 116.502
10\103 116.505 103fg val
17\175 116.571 175ffggg val
5/3 116.588 5- and 7-odd-limit minimax
11/10 116.591
11/6 116.596
11/7 116.617
7/6 116.641
7\72 116.667 72fg val
9/5 116.716 9-, 11- and 13-odd-limit minimax
11/8 116.755
18\185 116.757 185cffggg val
9/7 116.792
11\113 116.814 113fgg val
3/2 116.993 15-odd-limit minimax
4\41 117.073 Upper bound of 11- to 17-odd-limit,
and 17-limit 21-odd-limit diamond monotone
13/11 117.266
13/9 117.559
17/11 117.597
13/12 117.936
17/9 118.087
17/12 119.400
15/14 119.443
13/8 119.824
21/17 121.942
15/13 123.871
13/7 128.298

Benediction

Edo
generator 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
15/8 111.731
7/4 115.587
11/9 115.803
17/13 116.107
3\31 116.129 Lower bound of 11- to 17-odd-limit,
and 17-limit 21-odd-limit diamond monotone
5/4 116.241
15/11 116.441
13/8 116.455
17/16 116.501
7/5 116.502
10\103 116.505
13/7 116.509
13/10 116.511
13/12 116.536
13/11 116.547
17/14 116.567
17\175 116.571 175f val
17/10 116.581
17/12 116.583
17/11 116.586
5/3 116.588 5-, 7- and 15-odd-limit minimax
11/10 116.591
13/9 116.595 13-odd-limit minimax
11/6 116.596
15/13 116.598
11/7 116.617
7/6 116.641
17/9 116.642
21/17 116.642
17/15 116.666
7\72 116.667 Upper bound of 13- to 17-odd-limit,
and 17-limit 21-odd-limit diamond monotone
9/5 116.716 9- and 11-odd-limit minimax
11/8 116.755
18\185 116.757 185cfffgg val
9/7 116.792
11\113 116.814 113ffg val
3/2 116.993
4\41 117.073 41fg val, upper bound of 11-odd-limit diamond monotone
15/14 119.443

Manna

Edo
generator 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
15/8 111.731
7/4 115.587
11/9 115.803
17/13 116.107
3\31 116.129 31fg val, lower bound of 11-odd-limit diamond monotone
5/4 116.241
15/11 116.441
7/5 116.502
10\103 116.505 103ffgg val
17\175 116.571 175fffgg val
5/3 116.588 5- and 7-odd-limit minimax
11/10 116.591
11/6 116.596
11/7 116.617
7/6 116.641
7\72 116.667 Lower bound of 13- to 17-odd-limit,
and 17-limit 21-odd-limit diamond monotone
17/15 116.667
21/17 116.689
17/9 116.702
17/10 116.707
9/5 116.716 9- and 11-odd-limit minimax
15/13 116.725 15-odd-limit minimax
17/14 116.730
17/12 116.750
11/8 116.755
18\185 116.757 185cf val
13/10 116.760 13-odd-limit minimax
17/16 116.785
9/7 116.792
13/7 116.79254
13/9 116.79299
17/11 116.801
11\113 116.814
13/12 116.830
13/8 116.856
13/11 116.922
3/2 116.993
4\41 117.073 Upper bound of 11- to 17-odd-limit,
and 17-limit 21-odd-limit diamond monotone
15/14 119.443

Revelation

Edo
generator 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
15/8 111.731
13/10 113.553
2\21 114.286
13/11 114.555
11/10 115.000
5\52 115.385 52f val
11/7 115.536
11/8 115.543
7/4 115.587
15/11 115.797
11/6 115.938
3\31 116.129 11- to 15-odd-limit,
and 13-limit 21-odd-limit diamond monotone (singleton)
11/9 116.164 11-, 13- and 15-odd-limit minimax
5/4 116.241
7/5 116.502
5/3 116.588 5- and 7-odd-limit minimax
7/6 116.641
7\72 116.667 72ee val
9/5 116.716 9-odd-limit minimax
9/7 116.792
3/2 116.993
4\41 117.073 41ef val
13/9 117.559
13/12 117.936
15/14 119.443
13/8 119.824
15/13 123.871
13/7 128.298
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